Analysis of Metric Spaces

Explore infinity as a process and infinite sets (chapter two), define finite sets, and introduce set equivalence via bijections, setting the stage for topological and metric spaces.

Examine how infinite sets relate to proper subsets via 1-to-1 and onto functions. Compare finite, countable, and uncountable sets using sequences.

Analyze sequences and subsequences, define countable sets and unions and intersections, explore De Morgan's laws, and illustrate Cantor's theorem with the countability of rational numbers and multiple proofs.

Analyze the relationship between real and rational numbers, understand the completeness property, and learn how decimal expansions distinguish rationals from irrationals via repeating and terminating decimals.

Explore why the real numbers are uncountable, using decimal expansions, repeating decimals, and Cantor's diagonal argument, with connections to fractions and set theory.

Explore Cantor's diagonal argument proving real numbers are uncountable, inspect finite, countable, and uncountable sets, and introduce Cartesian products, equivalence relations, and invertible mappings in set theory.

Examine the paradox of the set of all sets, then introduce metric spaces defined by a distance function, including open neighborhoods and limit points.

Explore neighborhoods in metric spaces by defining radius-based balls, and identify limit points, interior points, and isolated points, while comparing open and closed sets in Euclidean space.

Explore interior points, open and closed sets, and limit points in metric spaces through neighborhoods, boundaries, and set complements.

Explore dense sets and perfect sets in metric spaces, using limit points and neighborhoods to contrast closed, discrete, and rationals in R.

Analyze closed, bounded, and compact sets within metric spaces, clarifying how these properties interact and framing compactness in the context of metric space analysis.

Examine how open and closed sets relate in metric spaces, proving open iff complement is closed, via interior points, limit points, and De Morgan's laws.

Explore the closure operation in metric spaces, linking closedness and limit points, and contrast open and closed sets through unions, intersections, and the role of compactness as analogy to boundedness.

Explore how compactness generalizes closed intervals in metric spaces via open covers and finite subcovers. Realize that compact sets are closed and finite point sets are compact.

Examine compactness in metric spaces, showing compact sets are closed, and that in Euclidean space closed and bounded subsets are compact. Show continuous images of compact sets remain compact.

Explore compactness through finite unions of compact sets, intersections (or empty), and open covers with finite subcovers, and discuss cartesian products, product versus box topology, Tychonoff's theorem, and perfect sets.

Explore the properties of compact and perfect sets in metric spaces, focusing on limit points, interior, closure, and the finite intersection property, including the uncountability of perfect sets.

Explore the real line through sequences and series, covering divergence and comparison tests, the geometric series, and Cauchy condensation, with a focus on convergence via bounded partial sums.

Explore compactness and open covers in metric spaces, apply De Morgan's laws, and construct the Cantor set in [0,1], illustrating measure zero and uncountability.

Examine connectedness in metric spaces via partitions into open and closed components and limit points, then introduce convergence of sequences in metric spaces using epsilon–N definitions.

Explore convergence and limit points of sequences in metric spaces using epsilon-neighborhood criteria, with complex-number examples, divergence cases, and the distinction between limits and limit points.

An exploration of completeness in metric spaces through Cauchy sequences, including the relation to compactness and completion, with examples from countable spaces, limit points, and subsequences.

Explore the Cauchy criterion for convergence in sequences and metric spaces, and compare it with monotone completeness, series, and standard tests such as divergence and comparison.

Examine two equivalent definitions of continuity between metric spaces: preimage of open sets and epsilon-delta conditions. Discuss implications for isometries, composition, and power series with root tests.